SpaceX’s February 6 launch of a Tesla Roadster into a Mars-crossing orbit briefly mesmerized the planet. The Tesla is now well on its way and not expected to undergo any further course changes. In other words, its trajectory is set.

And that raises an interesting question. What is this vehicle’s destiny? When will it pass close to Earth again? And how will it die—in a fiery impact with Earth, Mars, Venus, or even the sun?

Today we get an answer thanks to the work of Hanno Rein at the University of Toronto in Canada and colleagues. These guys have calculated the future trajectory of the car as far as is possible and say that over the next million years it has a 6 percent and 2.5 percent chance of colliding with Earth and Venus, respectively.

This kind of calculation is harder than it might initially seem. It’s easy to imagine that once the craft’s velocity is known, it is a simple matter to predict its trajectory for the rest of eternity given the relatively stable forces of gravity that will be acting on it.

Not so. Astronomers have long tried to determine the fate of near-Earth asteroids, comets, and even the planets themselves over the lifetime of the solar system, with limited success. The trouble is that the system is hugely complex. The trajectories of these objects are influenced by the gravitational attraction of many other objects, the positions of which have to be known precisely.

But it’s impossible to measure the velocity of all these objects exactly, so the simulations often produce wildly different predictions depending on the starting numbers.

(There are even phenomena such as the Yarkovsky effect to take into account. This is the momentum change generated by the emission of thermal photons, an effect that is complicated by the way objects rotate and the rate at which they cool down. For the Tesla, Rein and co calculate this to be small over the next 1,000 years compared with the gravitational nudges the vehicle will get from close approaches to planets.)

In just the same way, the precise parameters of the Tesla’s initial trajectory have a huge impact on its ultimate fate. A small change in these parameters can make a big difference in the outcome. In other words, the vehicle’s trajectory is chaotic.

That’s not unexpected. Astronomers see the same behaviour with near-Earth asteroids, and even with the planets themselves, when they simulate the future of these objects.

This makes it impossible to say exactly where the Tesla will be over time periods longer than a few hundred years. However, by running lots of simulations, each with a slightly different set of starting conditions, astronomers can see the various likely scenarios and work out how likely they are. This eventually reveals the probability of the Tesla smashing into Earth or Mars or Venus or anything else in the long term.

And that’s exactly what Rein and co have done. They performed several hundred simulations of the Tesla’s future trajectory around the sun over the next million years and beyond. In each one, they add a tiny random variation to the initial velocity of the vehicle and then see where it ends up.

The results make for interesting reading. First, the Tesla comes within the moon’s distance of Earth in 2091. That’s a dead cert.

After that, the simulated trajectories rapidly diverge. However, in none of them does the Tesla collide with Earth within the next 1,000 years.

Indeed, over the next million years, the number of collisions with Earth or other planets is small. This allows the team to estimate the probability of collision over this time scale. “We numerically calculate a collision probability of ≈ 6% and ≈ 2.5% with the Earth and Venus over one million years, respectively,” they say.

And that leads to a rough estimate that the vehicle’s likely life span will be a few tens of millions of years.

The ultimate fate of the spacecraft carrying the Tesla is still unknown. Much will depend on the orbital dynamics of the solar system itself. Astronomers have long known that asteroids often end up in orbits that resonate with the orbits of larger objects such as Jupiter and Earth. When this happens, they become much less likely to collide with the planets and generally end their lives colliding with the sun after a great deal of time.

The Tesla’s trajectory is a little different from that of most asteroids in that it originated from Earth. In that respect, it is more like the ejecta from an asteroid impact with our planet.

Just what will happen to the car is still up for grabs. Indeed, predicting it will require a larger number of simulations over much longer time scales. Or as Rein and co put it: “Much longer integrations are needed to quantify whether most of the remaining realizations would result in impacts with the terrestrial planets, or whether the Tesla can diffuse into strong resonances capable of driving it into the Sun.”

If anybody has a bit of spare time and a reasonably powerful supercomputer, then the answer is there for the taking. We’d love to know the result.

Ref: arxiv.org/abs/1802.04718 : The Random Walk of Cars and Their Collision Probabilities with Planets

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